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On the existence of solutions and one-step method for functional differential equations with parameters. (English) Zbl 0817.34006

Der Gegenstand der Arbeit ist die Lösung der Randwertaufgabe für die Volterrasche neutrale Funktional-Differentialgleichung der Form \[ x'(t)= f(t,x(.), x'(.),\lambda),\quad t\in I,\tag{1} \]
\[ x(a)= x_ p,\;L(x(.), \lambda)= \theta\in \mathbb{R}^ q,\tag{2} \] wobei \(f: I\times C^ 1(I, \mathbb{R}^ p)\times C(I,\mathbb{R}^ p)\times \mathbb{R}^ q\to \mathbb{R}^ p\), \(L: C^ 1(I, \mathbb{R}^ p)\times \mathbb{R}^ q\to \mathbb{R}^ q\), \(x_ p\in \mathbb{R}^ p\) gegeben sind und \(I= [a,b]\), \(a< b\). Die Aufgabe ist durch die Substitution \(y= x'\) äquivalent mit einer nichtlinearen Integralgleichung. Diese neue Aufgabe löst der Autor mittels Verwendung des Einschrittverfahrens für \(y\) kombiniert mit der Newtonschen Methode für \(\lambda\). Im weiteren wird die Konvergenz der erreichten Lösungsfolgen untersucht. Die erreichten Ergebnisse sind eine Erweiterung der Ergebnisse früherer Arbeiten des Autors [Math. Nachr. 71, 237-247 (1976; Zbl 0384.34046); Math. Nachr. 125, 7-28 (1986; Zbl 0587.34049); Computing 43, 343-359 (1990; Zbl 0689.65053)].

MSC:

65J99 Numerical analysis in abstract spaces
34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
65Q05 Numerical methods for functional equations (MSC2000)
35K05 Heat equation
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References:

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